CDS (II) 2023 General Knowledge · Question 4 of 120

Correct — D, increase to four times of its original value. Newton's law of universal gravitation says the pull between two bodies is F = GMm/r², where r is the distance between their centres. Neither mass changes here, so G, M and m stay put and the whole answer sits in the r² underneath. Replace r by r/2 and the denominator becomes (r/2)² = r²/4, which is a quarter of what it was; dividing by a quarter is multiplying by four. The force therefore rises to four times its original value. The general habit is worth fixing: because the distance is squared, a change in separation always bites twice as hard as the same change in a mass. Double the distance and the force falls to a quarter; treble it and the force falls to a ninth.

• (a)reduce to half of its original value — Gets both the direction and the size wrong. Bringing the bodies closer strengthens the attraction, and even the strengthening would not be a simple factor of two because the distance is squared.
• (b)increase to two times of its original value — The right direction with the wrong exponent. Doubling would be the answer if the force went as 1/r; it goes as 1/r², so the factor is 2² = 4.
• (c)remain the same as the original value — Would be true only if the force did not depend on separation at all. It does — distance is the one thing this question changes, and the force follows it.

Newton's law of universal gravitation states that every particle of matter attracts every other with a force proportional to the product of their masses and inversely proportional to the square of the distance between them: F = GMm/r². The constant G is about 6·67 × 10⁻¹¹ N m² kg⁻², the same everywhere in the universe, and it is what makes gravitation utterly negligible between everyday objects and dominant between astronomical ones. The force is always attractive, acts along the line joining the two bodies, and forms an action–reaction pair — the Moon pulls on the Earth exactly as hard as the Earth pulls on the Moon.

This is a one-line inverse-square drill dressed up as an astronomy question, and the mention of the Moon is decoration. The safe method is to write the ratio rather than the formula: F_new/F_old = (r_old/r_new)², so halving the separation gives a ratio of 2² = 4. Doing it this way stops the two standard slips — forgetting to square, and getting the direction backwards. Note that the physical scenario is impossible in a stable sense: at half its distance the Moon would be inside the Roche limit region for tidal stress and the tides on Earth would be enormously stronger, since tidal effect falls off even faster, as 1/r³. The question asks only about the force between them, so none of that is needed for the mark, but it is why the item cannot be reasoned from intuition about orbits.

• The gravitational force between two bodies is F = GMm/r², with r measured centre to centre.
• The universal gravitational constant G is about 6·67 × 10⁻¹¹ N m² kg⁻² and does not change with place, medium or time.
• Halving the separation multiplies the force by four; doubling it divides the force by four.
• Gravitational force is always attractive and always acts along the line joining the two bodies, as an equal and opposite pair.
• The mean Earth–Moon distance is about 384,400 km, and the same inverse-square law is what holds the Moon in orbit and what raises the ocean tides.

• Forgetting to square the distance factor, which turns a factor of four into a factor of two.
• Reversing the direction — closer means a stronger pull, not a weaker one.
• Applying the same reasoning to tides, which weaken as the cube of the distance rather than the square.

Almost always as a proportional-change one-liner: change a mass, or the separation, or both, and state what happens to the force.

• practice — not a real PYQ

  If the masses of two bodies are each doubled and the distance between them is also doubled, the gravitational force between them will

  • (a)remain the same
  • (b)become half
  • (c)become four times
  • (d)become sixteen times

  Answer(a) remain the same — the product of the masses goes up by four and the square of the distance also goes up by four, so the two changes cancel exactly.

• practice — not a real PYQ

  The value of the universal gravitational constant G

  • (a)is greater at the poles than at the equator
  • (b)decreases with height above the Earth's surface
  • (c)is the same everywhere in the universe
  • (d)depends on the medium between the two bodies

  Answer(c) is the same everywhere in the universe — G is a universal constant. It is g, the acceleration due to gravity, that varies with latitude, altitude and depth.